# Amenability by Alan L. T. Paterson

By Alan L. T. Paterson

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12. Suppose α > −1, β > −1, and 1 < p < ∞. Then (Apα )∗ = Aqβ (with equivalent norms) under the integral pairing f, g where γ = Bn f (z) g(z) dvγ (z), 1 1 + = 1, p q γ= f ∈ Apα , g ∈ Aqβ , α β + . p q 48 2 Bergman Spaces Proof. If g ∈ Aqβ and F (f ) = f, g γ = cγ Bn (1 − |z|2 )α/p f (z)(1 − |z|2 )β/q g(z) dv(z), f ∈ Apα , it follows from H¨older’s inequality that F is a bounded linear functional on Apα with F ≤ C g q,β , where C is a positive constant depending on cα , cβ , and cγ . Conversely, if F is a bounded linear functional on Apα , then according to the Hahn-Banach extension theorem, F can be extended (without increasing its norm) to a bounded linear functional on Lp (Bn , dvα ).

15. We can use the above lemma to compare the behavior of Rs,t and Rβ,t . In fact, in a very broad sense, Rs,t f and Rβ,t f are comparable for any holomorphic function f . To see this, we write N Ck (1 − z, w )k , h( z, w ) = k=0 where Ck are constants. 14, is the same as Rs,t 1 = (1 − z, w )n+1+β N Ck Rβ+t−k,k Rβ,t k=0 1 . (1 − z, w )n+1+β Differentiating with respect to w, we obtain N Rs,t = C0 Rβ,t + Ck Rβ+t−k,k Rβ,t . k=1 It is easy to see that C0 = 0. Also, for each 1 ≤ k ≤ N , the function Rβ+t−k,k Rβ,t f is a k-th integral of Rβ,t f , and so is more regular than Rβ,t f .

20. Suppose a ∈ Bn and f is a function in Bn . Show that f ◦ ϕa = f if and only if f = g ◦ ϕma , where g is an even function in Bn ; similarly, f ◦ ϕa = −f if and only if f = g ◦ ϕma , where g is an odd function in Bn . 21. If Ja (z) is the complex Jacobian determinant of ϕa at z ∈ Bn , show that Ja (z) = (−1)n (1 − |a|2 )(n+1)/2 . 22. Show that Cn f (z) dv(z) =n |z|2n dσ(ζ) Sn C f (ζw) dA(w) |w|2 whenever the integrals converge. 23. Suppose f and g are twice continuously differentiable functions with compact support in Bn .